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We study the spin-$1/2$ Heisenberg model on the square lattice with the firstand second nearest-neighbor antiferromagnetic couplings $J_1$, $J_2$, as wellas the three-spin scalar chiral coupling $J_{\chi}$. Using density matrixrenormalization group calculations, we obtain a quantum phase diagram of thissystem for $0 \leq J_2/J_1 \leq 1.0$ and $0 \leq J_{\chi}/J_1 \leq 1.5$. Weidentify the N\'eel and stripe magnetic order phase at small $J_{\chi}$coupling. With growing $J_{\chi}$, we identify the emergent chiral spin liquid(CSL) phase characterized by the quantized spin Chern number $C = 1/2$ andentanglement spectrum with the quasidegenerate group of levels agreeing withchiral SU(2)$_1$ conformal field theory, which is an analog of the $\nu = 1/2$Laughlin state in spin system. In the vicinity of the N\'eel and CSL phaseboundary, our numerical results do not find evidence to support the phasecoexistence of N\'eel order and topological order that was conjectured bymean-field calculations. In the larger $J_2$ and $J_{\chi}$ coupling regime,the entanglement spectrum of the ground state also exhibits the chiralquasidegeneracy consistent with a CSL, but the adiabatic flux insertionsimulations fail to obtain the quantized Chern number. By analyzing thefinite-size scaling of magnetic order parameter, we find the vanished magneticorder suggesting a magnetic disorder phase, whose nature needs further studies.Different from the spin-$1$ $J_1$-$J_2$-$J_\chi$ model, we do not find thecoexistent stripe magnetic order and topological order. We also investigate the$J_{\chi}$ dominant regime and find a strong tendency of the system to developa dimer order rather than the chiral spin magnetic order observed in thespin-$1$ model.

Chiral spin liquid and quantum phase diagram of spin-$1/2$ $J_1$-$J_2$-$J_{\chi}$ model on the square lattice